Normality-like properties, paraconvexity and selections.

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Abstract

In 1956, E. Michael proved his famous convex-valued selection theorems for l.s.c. mappings
de ned on spaces with higher separation axioms (paracompact, collectionwise
normal, normal and countably paracompact, normal, and perfectly normal), [39]. In
1959, he generalized the convex-valued selection theorem for mappings de ned on paracompact
spaces by replacing \convexity" with \ -paraconvexity", for some xed constant
0 < 1 (see, [42]). In 1993, P.V. Semenov generalized this result by replacing
with some continuous function f : (0;1) ! [0; 1) (functional paraconvexity) satisfying
a certain property called (PS), [63]. In this thesis, we demonstrate that the classical
Michael selection theorem for l.s.c. mappings with a collectionwise normal domain can
be reduced only to compact-valued mappings modulo Dowker's extension theorem for
such spaces. The idea used to achieve this reduction is also applied to get a simple
direct proof of that selection theorem of Michael's. Some other possible applications
are demonstrated as well. We also demonstrate that the -paraconvex-valued and the
functionally-paraconvex valued selection theorems remain true for C 0
(Y )-valued mappings
de ned on -collectionwise normal spaces, where is an in nite cardinal number.
Finally, we prove that these theorems remain true for C (Y )-valued mappings de ned
on -PF-normal spaces; and we provide a general approach to such selection theorems.